🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{a^3 \cdot a^{-5} \cdot b^{-4}}{(a \cdot a^{-4} \cdot b^2)^3}$$ so that all exponents are positive. 2. **Recall exponent ru

1. **State the problem:** Simplify the expression $$\frac{(m^{-1})^{-1} \cdot (m^2)^{-5}}{m^{-1} \cdot m^5}$$. 2. **Apply the power of a power rule:** $$(a^b)^c = a^{bc}$$.

1. **State the problem:** Simplify the expression $$\frac{(2c^6)(4c)}{8c^3}$$ and express the answer with positive exponents. 2. **Write the expression:** $$\frac{(2c^6)(4c)}{8c^3}

1. **State the problem:** Solve the quadratic equation $$5x^2 = 3 - x$$ and find the values of $x$ correct to 2 decimal places. 2. **Rewrite the equation:** Move all terms to one s

1. **State the problem:** Solve the quadratic equation $$2x^2 + 3x = 10$$ for $x$, with answers correct to 2 decimal places. 2. **Rewrite the equation:** Move all terms to one side

1. **State the problem:** Simplify the expression $-4(2x-5y)$. 2. **Formula and rule:** Use the distributive property: $a(b+c) = ab + ac$.

1. **State the problem:** Simplify the expression $-4(2x-5y)$. 2. **Formula and rule:** Use the distributive property: $a(b+c) = ab + ac$.

1. **State the problem:** An item is on sale for 45% of its original price. The original price is 35. 2. **Formula:** Sale price = Original price \times Percentage of original pric

1. **State the problem:** An item originally priced at 35 is on sale for 45% of the original price. We need to find the sale price. 2. **Formula used:** Sale price = Original price

1. **State the problem:** We know that 28,100 employees represent 18.2% of the total employees of the company. We need to find the total number of employees. 2. **Formula used:** T

1. **State the problem:** We need to find the minimum price for a dozen eggs if the total cost of four dozen eggs plus a half gallon of milk costing 2.97 exceeds 10.

1. **State the problem:** An automobile salesman makes 100 plus a 2% commission on the sale price of a car. We want to find the minimum sale price of a car so that the salesman mak

1. Problema: Calcula simplificando durante el cálculo siempre que sea posible la expresión a) $\frac{3}{5} + \frac{2}{3} \cdot \frac{1}{5} + \frac{1}{3}$. 2. Fórmulas y reglas impo

1. **State the problem:** Given $|a|=5$, $|b|=20$, and $|a-b|=15.5$, find $|a+b|$. 2. **Recall the formula:** For any vectors or numbers $a$ and $b$, the identity

1. The problem is to verify if the given evaluated expressions are correct for the specified variable values. 2. We will check each expression step-by-step by substituting the give

1. **Problem statement:** Find an explicit formula for the sequence defined by $a_1 = -4$ and the recursive relation $a_n = a_{n-1} + 9$ for $n \geq 2$. 2. **Understanding the prob

1. **State the problem:** Evaluate the expression $$2 + 3t$$ for $$t = 10$$. 2. **Write the formula:** The expression is $$2 + 3t$$.

1. **State the problem:** We need to find the equation of a circle given its center and radius from the graph. 2. **Recall the formula for a circle's equation:** The standard form

1. **Stating the problem:** We need to verify if the answers given for the algebra questions are correct. 2. **Definition of a variable:** A variable is an unknown number or symbol

1. **State the problem:** We need to find the equation of a circle centered at the point $(5, 1)$ with a radius of $2$ units. 2. **Recall the formula for a circle's equation:** The

1. **State the problem:** We need to find the equation of a circle given its center and radius. 2. **Recall the formula:** The standard form of a circle's equation with center $(h,